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8 %%\cite{proakis2004contemporary}
9 %%--01-- pointer or label to related works
11 %define document class
12 \documentclass[a4paper, 12pt]{article}
14 %packages for language and input
15 \usepackage[english]{babel}
16 \usepackage[T1]{fontenc}
17 \usepackage[latin1]{inputenc}
25 \usepackage[pdfpagelabels]{hyperref}
26 \usepackage{longtable}
27 \usepackage{float} %For restylefloat
30 %isabelle relevant packages
31 \usepackage{isabelle,isabellesym}
34 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
35 \def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
37 %----------// BEGIN DOCUMENT \\----------%
41 %----------// INFO SETUP \\----------%
45 \bf Interactive Course Material for Signal Processing based on Isabelle/\isac\\~\\
47 \sisac-Team in Cooperation with \\~\\
48 Institute for Software Technology\\
49 Institute of Signal Processing and Speech Communication\\~\\
50 Graz University of Technology\\
53 Supervisor: Univ.-Prof. Dipl.-Ing. Dr.techn. Franz Wotawa
57 \author{Jan Simon Rocnik\\\href{mailto:student.tugraz.at}{\tt jan.rocnik@student.tugraz.at}}
61 %----------// TITLE PAGE \\----------%
65 \thispagestyle{empty}\end{titlepage}
68 %----------// THANKS \\----------%
74 \emph{Dr.techn. Walther Neuper}\\
75 \emph{Dipl.-Ing. Bernhard Geiger}
80 %----------// ABSTRACT \\----------%
83 The Baccalaureate Thesis creates interactive course material for Signal Processing based on the experimental math assistant Isabelle and provides it within {\sisac} (Isabelle for Calculations).
84 \par The content of the course material is defined together with the Signal Processing and Speech Communication Laboratory (SPSC Lab) of Graz University of Technology (TUG). The content is planned to be used in specific lectures and labs of the SPSC and thus is thoroughly concerned with underlying mathematical and physical theory.
85 One challenge of this thesis is, that theory is not yet mechanized in Computer Theorem Provers (CTP); so this thesis will provide preliminary definitions in so-called \emph{theories} of the CTP Isabelle and theorems without proofs.
86 \par Another callenge is the implementation of interactive courses: this is done within the educational math assistant Isabelle/{\sisac}, which is under development at TU Graz. The present state of {\sisac{}} happens to provide the {\em first} occasion for authoring by a non-member of the {\sisac}-developer team. So this challenge involves alpha-testing of the underlying \emph{CTP-based programming language}, because error messages are still not user-friendly and need frequent contact with {\sisac}-developers.
87 So the practical outcome of this thesis is twofold:
89 \item Interactive course material hopefully useful in education within the SPSC Lab and within STEOP, the introductory orientation phase at TUG, as a preview for students in Telematics on later application of math knowledge introduced in the first semester and
90 \item A detailed description of technicalities in programming implemented as an interactive Isabelle/Isar theory, providing future programmers with guidelines and {\sisac}-developers with feedback in usability of the CTP-based program language.
92 \end{abstract}\clearpage
94 %----------// T O C \\----------%
97 This thesis is structured into a generell part describing the math fundamentals, a practical part including the work on \cal{ISAC} and finally the management part overviewing the work process.
100 \pagenumbering{arabic}
103 %----------// PART-1 \\----------%
105 \part{Project Fundamentals}
107 \section{Introduction}
108 The motivation to this thesis mainly takes it source from the feeling of understanding difficult signal processing tasks and the will to help others to get this feeling to.
109 \par Signal Processing requieres a huge range of mathematic knowledge as well as a feeling for simplification and number tricks but even though this fact, the operations themself are no higher ones. The main task is to understand. Aside this description we think of the classic math ideas and techniques, consisting of predefined formulas, notations and forumularsations we learn.
110 \par Mathematics mechanized in Computer Theorem Provers (\emph{CTP}) has (almost) a problem with traditional mathematical notations (predicate calculus) for axioms, definitions, lemmas, theorems as a computer programm or script is not able to interpret every greek or latin letter and every greek, latin or whatever calculations symbol. Also if we would be able to handle thehse symbols we would have a problem to interpret them correctly. In different problems, symbols and letters have different meanings and ask for different ways to get through. Exclusive from the input, also the output can be a problem. We are familar with a specified notations and style taught in university but a computer programm has no knowledge of the form probved by a professor and the maschines themselve also have not yet the possibilities to print every symbol (correct) Recent developments provide proofs in a humand readable format but according to the fact that there is no mony for good working formel editors yet, the style is one thing we have to live with.
111 \par This thesis tries to \emph{connect} these two worlds and is one of the first guidelines to implement problem classes in {\sisac}. For others see related works in section \ref{sec:related}.
112 The major challenge of the practical part, of this thesis, is, that "`connecting the two worlds"' involves programming in a CTP-based programming language which is in a very early state of prototyping. There is no concrete experience data ready to grep.
114 \subsection{Mechanization of Mathematics}
115 A problem behind is the mechanization of mathematic theories in CTP-bases languages. There is still a hugh gap between these theories and this what we call an applications - in Example Signal Processing. Until we are not able to fill this gap we have to live with it but first have a look on the meaning of this statement:
116 \par Mechanized math starts from mathematical models and \emph{hopefully} proceeds to match physics. Academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modelling and formalization. The process from a physical observance to a mathematical theory is unavoidable bound of setting up a big collection of standards, rules, definition but also exceptions. These are the things making mechanization that difficult.
117 \par A computer or a CTP-System builds on programms witth predefined logical ruels and does not know any mathematical trick or recipe to walk around difficult expressions. For such a system the only possibility is to work through its known definitions vulgo theories and stops if none of these fits. Specified on Signal Processing or any other application it is often possible to walk through by doing simple creases. This creases are in generell based on simple math operatiopms but the challange is to teach the machine \emph{all}\footnote{Its pride to call it \emph{all}.} of them. Unfortunataly the goal of CTP Isabelle is to reach a high level of \emph{all} but it in real it will still be a survey of knowledge which links to other knowledge and {\sisac{}} a trainer and helper but no human compensating calulator.
118 \par {\sisac{}} itselfs aims to adds an \emph{application} axis (formal specifications of problems outof topics from Signal Processing, etc.) and an \emph{algorithmic} axis to the \emph{deductive} axis of physical knowledge. The result is a three-dimensional universe of mathematics.
120 \subsection{Goals of the Thesis}
121 Imagine a piece of software would be able to support you by understanding every problem class, upcoming in the first years attending university - wouldn't it be great?
122 \par {\sisac{}} tries to do that, but the current state of the art is miles away from this goal and a single implementation of a problem is not enough to cahnge this circumstamce. Through this fact it is all the more essential to try, test, research and document the implementation of problem classes from "`real world"' applications. Responding to the abstract at the begin of this document the thesis has two folds; on the one hand certainly to provide interactiv course material for Signal Processing (which means to implement a single problem provided by the Institute of Signal Processing and Speech Communication (SPSC); follow up Calulcations), and to extract experience data respectively help the {\sisac{}}-team by setting up a detailed description of technicalities hacking {\sisac{}} on the other hand.
123 \par Another goal is to demonstrate the power and attractivity of {\sisac}.
125 \section{Mechanization of Signal Processing Problems}
126 \subsection{Relevant Knowledge available in Isabelle}
127 Isabelle is developed now for a long time and so we are able to access a huge range of theories and usefull snipsets. The main problem according this snipsets is that isabelle still is a theorem proofer and not an algebra system. But due the work of the {\sisac}-development team there are already also many calculation examples provided.
128 \par The SPSC provided a list of problems which are often done wrong or are missunderstood by studentsin term of the problem classes. Out of these tasks we tried to extract the core operations and looked up which parts are already implemented or usefull. The provided problems are:
130 \item Fourier-Transformation
132 \item Inverse z-Transformation and partial fraction decomposition
133 \item Indextransformation
135 Following the collection and evaluation of core operations collated with isabelle:
137 \paragraph{example FFT}, describe in detail !!!!
139 ? different meaning: FFT in Maple
141 gap between what is available and what is required (@)!
143 traditional notation ?
145 \subsection{Relevant Knowledge available in isac}
148 specifications (``application axis'') and methods (``algorithmic axis'')
150 partial fractions, cancellation of multivariate rational terms, ...
152 \subsection{Survey: Requiered Knowledge and Selected Problem(s)}
153 Following tables are showing the expected development effort for speciefic problems. The values are only very inaccurately approximations of the real work, but needed as a basis for descieding with which problem to start:
157 \begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}
158 requirements & comments &effort\\ \hline\hline
159 solving Intrgrals & simple via propertie table & 20\\
160 & \emph{real} & MT\\ \hline
161 transformation table & simple transform & 20\\ \hline
162 visualisation & backend & 10\\ \hline
163 example collection & with explanations & 20\\ \hline\hline
164 \multicolumn{2}{c|}{} & 70-80\\
167 \caption{Fourier-Transformation development effort}
172 \begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}
173 requirements & comments &effort\\ \hline\hline
174 simplify rationals & \sisac & 0\\ \hline
175 define $\sum\limits_{i=0}^{n}i$ & partly \sisac & 10\\ \hline
176 simplify sum & termorder & 10\\
177 & simplify rules & 20\\
178 & use simplify rationals& 0\\ \hline
179 index adjustments & with unit step & 10\\ \hline
180 example collection & with explanations & 20\\ \hline\hline
181 \multicolumn{2}{c|}{} & 70-90\\
184 \caption{Convolution Operations development effort}
189 \begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}
190 requirements & comments &effort\\ \hline\hline
191 solve for part.fract. & \sisac: degree 2 & 0\\
192 & complex nomminators & 30\\
193 & degree > 2 & MT\\ \hline
194 simplify polynomial & \sisac & 0\\
195 simplify rational & \sisac & 0\\ \hline
196 partial fraction & degree 2, & 20\\
197 decomposition & specification, method& 30\\ \hline
198 ${\cal Z}^{-1}$ table & explanations, figures& 20\\ \hline
199 example collection & with explanations & 20\\ \hline\hline
200 \multicolumn{2}{c|}{} & 90-120\\
204 \caption{Z-Transformation development effort}
207 As conclusion of the summerized efforts it is evident that only one topic can be tried to realized as a baccalaureate thesis. In accord with Dr. Neuper we decided after some practical tests to start with the implementation of the (Inverse) Z-Transformation. The Reason is that this topic can mostly be done with knowledge which was already tried to be mechanized in {\sisac}.
209 \subsection{Formalization of missing knowledge in Isabelle}
212 axiomatization ... where ... and
214 \subsection{Notes on Problems with Traditional Notation}
215 Due the thesis work we discorvers severell problems of traditional notations.
219 f x = why not f(x) ?!?!
223 terms are not full simplified in traditional notations, in isac we have to simplify them complete to check weather results are compatible or not. in e.g. the solutions of an second order linear equation is an rational in isac but in tradition we keep fractions as long as possible and as long as they are 'beautiful' (1/8, 5/16,...)
227 %----------// PART 2 \\----------%
229 \part{Implementation}
231 %\section{Implementation of Certain SP Problems}
232 %tell why only choosen one problem given by geiger
234 %\subsection{Formal Specification of Problems}
237 %\subsection{Methods Solving the Problems}
240 %\subsection{Integration of Subproblems available in isac}
243 %\subsection{Examples and Multimedia Content}
247 \input{../../../test/Tools/isac/ADDTESTS/course/SignalProcess/document/Build_Inverse_Z_Transform}
251 %----------// PART 3 \\----------%
253 \part{Project Management}
255 \section{Milestones for the Project}
256 Die Planung des Projekts teilt sich in folgende Iterationen:
258 \item \textbf{Sammeln von Informationen "uber Themengebiete und deren Realisierbarkeit } (29.06. -- 27.07.)
259 identify problems relevant for certain SP lectures
261 estimate chances to realized them within the scope of this thesis
263 order for implementing the problems negotiated with lecturers
266 \item \textbf{1. Prsentation - Auswhlen der realisierbaren Themengebiete} (27.07.)
267 \item \textbf{Ausarbeiten der Aufgaben in \isac} (01.09. -- 11.11.)
268 \item \textbf{Dokumentation der Aufgaben} (14.11. -- 02.12.)
269 \item \textbf{Ausarbeitung in Latex, Bakkarbeit} (05.12. -- todo)
270 \item \textbf{2. Prsentation - Abschluss der Arbeit} (todo)
273 \section{Beschreibung der Meilensteine}\label{ms-desc}
275 \section{Bericht zum Projektverlauf}
277 \section{Abschliesende Bemerkungen}
282 %----------// RELATED \\-----------%
284 \section{Related Work and Open Questions \label{sec:related}}
286 See ``introduction'': This thesis tries to connect these two worlds ... this trial is one of the first; others see related work
289 %----------// BIB \\-----------%
291 \renewcommand{\refname}{\section{Sources}} % Using "Sources" as the title of the section
292 \bibliographystyle{alpha}
293 \bibliography{references}
296 %----------// APPENDIX \\-----------%
300 %----------// WORK TIME \\-----------%
302 \section{Stundenliste}
304 \begin{longtable}[h]{l p{6.5cm} c c r}
305 {\bf Date} & {\bf Description} & {\bf Begin} & {\bf End} & {\bf Dur.}\\
308 29.06.2011 & Treffen mit Geiger und Neuper & 15:00 & 17:30 & 2,50\\
309 02.07.2011 & Beispielaufbereitung (Bsp. Geiger Mail) & 20:00 & 21:30 & 1,50\\
310 03.07.2011 & Beispielaufbereitung, Vorraussetzungsausw. & 21:00 & 22:45 & 1,75\\
311 05.07.2011 & Treffen mit Neuper, Informationsaustausch & 10:00 & 13:00 & 3,00\\
312 06.07.2011 & Isabelle Installation & 20:00 & 22:30 & 2,50\\
313 07.07.2011 & Treffen mit Neuper, Präsentationsvorbereitung & 14:45 & 16:15 & 1,50\\
314 18.07.2011 & Präsentationsvorbereitung - Struktur & 14:15 & 16:00 & 1,75\\
315 19.07.2011 & Präsentationsvorbereitung - Inhalt & 07:20 & 09:20 & 2,00\\
316 19.07.2011 & Treffen mit Neuper & 10:00 & 12:00 & 2,00\\
317 21.07.2011 & HG Fehlersuche, Latex Ausarbeitung & 11:10 & 14:00 & 2,83\\
318 22.07.2011 & Treffen mit Neuper & 10:00 & 12:00 & 2,00\\
319 23.07.2011 & Berechnungen in Latex fertigstellen & 13:45 & 16:30 & 2,75\\
320 24.07.2011 & Präsentation fertigstellen & 20:10 & 20:40 & 0,50\\
321 25.07.2011 & Treffen mit Neuper, Präsentation \& erste Tests & 15:15 & 17:55 & 2,67\\
322 26.07.2011 & Test\_Complex.thy erarbeiten & 10:45 & 12:10 & 1,42\\
323 27.07.2011 & present-1 mit Neuper, Geiger & 10:00 & 12:00 & 2,00\\
325 02.09.2011 & Treffen mit Neuper, Vorlage Bakk-Arbeit & 08:30 & 10:20 & 1,83\\
326 05.09.2011 & Treffen mit Neuper, Beginn Partialbruchzerlegung & 09:30 & 12:45 & 3,25\\
327 05.09.2011 & Partialbruchzerlegung & 17:10 & 18:30 & 1,33\\
328 06.09.2011 & Dokumentation Partialbruchzerlegung & 10:00 & 13:15 & 3,25\\
329 07.09.2011 & Treffen mit Neuper, Einführung Programmierung & 10:00 & 12:50 & 2,83\\
330 08.09.2011 & Latex Umgebung einrichten - Theory export & 19:00 & 22:45 & 3,75\\
331 09.09.2011 & Latex Umgebung einrichten - Makefile & 11:40 & 15:00 & 3,33\\
332 10.09.2011 & Treffen mit Neuper, HG Fehler, Skript Inv.-Z-Transf. & 10:00 & 12:00 & 2,00\\
333 14.09.2011 & Skript Inv.-Z-Transf Prgrammierung & 09:10 & 12:25 & 3,25\\
334 16.09.2011 & Informationssammlung Summen & 13:15 & 16:00 & 2,75\\
335 19.09.2011 & Programmierübung & 10:00 & 13:10 & 3,17\\
336 20.09.2011 & Trefffen mit Neuper, Unterstützung bei Program. & 15:30 & 18:10 & 2,67\\
337 23.09.2011 & Neukonfiguration IsaMakefile & 13:00 & 14:30 & 1,50\\
338 23.09.2011 & Treffen Neuper, Programmierung Build\_Inverse\_Z & 14:30 & 17:30 & 3,00\\
339 26.09.2011 & Skript Partialbruchzerlegung - getArgument & 13:30 & 16:15 & 2,75\\
340 27.09.2011 & Treffen mit Neuper, HG Fehler & 09:00 & 12:20 & 3,33\\
341 28.09.2011 & Treffen mit Neuper, Dateiumstrukturierung & 10:00 & 12:30 & 2,50\\
342 01.10.2011 & Testen & 10:00 & 11:00 & 1,00\\
343 02.10.2011 & Fehlersuche & 15:00 & 16:10 & 1,17\\
344 06.10.2011 & Treffen mit Neuper & 15:00 & 17:50 & 2,83\\
345 07.10.2011 & Treffen mit Neuper, Programmbesprechung & 15:00 & 16:50 & 1,83\\
346 09.10.2011 & Bakk. Arbeit & 16:30 & 18:45 & 2,25\\
347 11.10.2011 & Treffen mit Neuper, Programmbespr., Abstract & 14:10 & 17:10 & 3,00
351 \section{Calculations}